Wavenumber lock-in and spatial parametric resonance in convection

Abstract

Analysis of convection driven by spatial periodic heating shows a wealth of responses due to Prandtl number dependence. The primary convection is delineated by the heating wavenumber and the secondary convection is characterized by the critical wavenumber established by an instability process. The resulting two-wavenumber dynamical system involves spatial parametric resonance leading to the wavenumber lock-in and the Rayleigh–Bénard (RB) instability producing unlocked states. Heating conditions leading to the co-existence of both types of states have been identified. Transition from the locked to the unlocked states produces states with a wide range of wavelengths and diverse patterns of movement. The possible secondary states are driven by a competition between the RB mechanism and the spatial parametric resonance. The relative strengths of these mechanisms change with Prandtl number $Pr$ resulting in four types of system responses. In the type A response occurring for $Pr>0.4$ , parametric resonance dominates for the heating wavenumber $\alpha < 4.5$ resulting in the pattern lock-in between the primary and secondary convections whereas RB occurs for $\alpha > 10$ . There is a wealth of possible convection patterns in the in-between zone where a small change of $\alpha$ results in a major change of flow pattern. In the type B response, which occurs for $0.19< Pr< 0.4$ , the RB effect dominates eliminating the lock-in. The type C response, which occurs for $0.08 < Pr< 0.19$ , is similar to type A but stronger spatial modulation extends the range of dominance of parametric resonance up to $\alpha =7$ while the RB effect dominates for $\alpha >10$ as in the type A response. A wealth of possible patterns occurs in the transition zone ( $7<\alpha <10$ ). In the type D response, which occurs for $Pr<0.08$ , the strong spatial modulation results in the formation of two separate critical stability curves, one resulting from the dominance of the spatial parametric resonance leading to the lock-in effect, and the other one corresponding to the dominance of the RB effect producing unlocked states. No continuous transition between both states can occur. Conditions where both distinct states can arise simultaneously were identified. Morphing between different patterns of secondary convection may occur in response to small changes in the heating pattern in types A and C, while such processes are not possible in types B and D.